Hey, it's been migrated! Congratulations! :) and meanwhile my earlier comment has been deleted in the rush. The second option is also possible if you consider "participant" in "participant ratings" functioning like an attributive noun.
– Mari-Lou AMar 4 at 18:53

Figures showing the distribution of the participant's ratings on the math tasks.

In the above sentence there is only one participant. For example: "The participant rated five of the tasks as 'hard' and five of the tasks as 'easy'."

In both of the above two examples possession is clearly stated with the presence of the apostrophe. The ratings are attributed to specific people who are the participants.

Figures showing the distribution of the participant ratings on the math tasks.

This sentence is a little different in that 'participant' is used as an adjective of 'ratings'. Normally 'participant' is a noun, but here it is used to describe the 'ratings'. "Page one lists participant ratings while page two lists spectator ratings." This post described how to use nouns as adjectives well:
https://www.englishclub.com/grammar/nouns-adjective.htm

I am assuming that there are many participants, and each has, or may have, submitted or generated a rating (or several). If that assumption is wrong, then the first is incorrect, and the third is correct.

If a word forms it plural in the usual way, adding -s (or similar), the possessive (or genitive) that would normally be formed with -'s is instead just formed with -'. Thus, if there are multiple participants who each gave rise to one or more ratings, there are many ratings originating in many participants, so both rating and participant should be plural, and participant should be possessive:

Figures showing the distribution of the participants' ratings on the math tasks

On the other hand, if you consider the fact a rating came from a participant to be an attribute of the rating, participant can be an attributive noun - which should be singular[1]:

Figures showing the distribution of the participant ratings on the math tasks

Whether the latter example is actually appropriate in your case depend on exactly what you are trying to say. It's grammatical and makes sense, but whether it's right depends on the actual underlying situation.

[1]: Attributive nouns are, I believe, grammatically singular even when the noun itself is one that is normally always plural - for example "trouser press".